← #7623201
Subgroups — #7623201
Order 512.
2512 conjugacy classes of subgroups.
The subgroup lattice has 220 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#3415 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #60647 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #33720 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | #33717 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #33713 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×#2246 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C4×#504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#576 | — | 2 | 2 | no | — |
| 9 | C2×#576 | C22 | 1 | 4 | yes | — |
| 10 | #3498 | — | 2 | 2 | no | — |
| 11 | #3415 | — | 2 | 32 | no | — |
| 12 | C2×#509 | C22 | 1 | 24 | yes | complemented |
| 13 | C2×#508 | C22 | 1 | 12 | yes | complemented |
| 14 | C22×D16 | C22 | 1 | 12 | yes | complemented |
| 15 | C2×#504 | C22 | 1 | 2 | yes | — |
| 16 | #3155 | — | 2 | 6 | no | — |
| 17 | #3155 | C22 | 1 | 12 | yes | complemented |
| 18 | #3151 | — | 2 | 6 | no | — |
| 19 | #3147 | C22 | 1 | 6 | yes | complemented |
| 20 | C2×#493 | C22 | 1 | 6 | yes | complemented |
| 21 | C4×#131 | C22 | 1 | 1 | yes | characteristic |
| 22 | #2250 | C22 | 1 | 24 | yes | complemented |
| 23 | #2249 | C22 | 1 | 12 | yes | complemented |
| 24 | #2248 | C22 | 1 | 6 | yes | complemented |
| 25 | #2247 | C22 | 1 | 1 | yes | characteristic, complemented |
| 26 | #2246 | C22 | 1 | 3 | yes | complemented |
| 27 | #2246 | C22 | 1 | 6 | yes | complemented |
| 28 | #2246 | C22 | 1 | 12 | yes | complemented |
| 29 | #2108 | C22 | 1 | 6 | yes | complemented |
| 30 | C4×M32 | C22 | 1 | 3 | yes | complemented |
| 31 | C2×C4×C16 | C22 | 1 | 3 | yes | complemented |
| 32 | C2×H(5,𝔽2) | — | 2 | 4 | no | — |
| 33 | #576 | — | 2 | 32 | no | — |
| 34 | #576 | — | 4 | 16 | no | — |
| 35 | C2×AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 36 | C2×AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 37 | C2×#135 | — | 2 | 6 | no | — |
| 38 | C2×#135 | C23 | 1 | 12 | yes | — |
| 39 | C22×D8 | — | 2 | 6 | no | — |
| 40 | C22×D8 | C23 | 1 | 12 | yes | — |
| 41 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 42 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 43 | #550 | — | 4 | 4 | no | — |
| 44 | D42 | — | 4 | 12 | no | — |
| 45 | #535 | — | 2 | 12 | no | — |
| 46 | #532 | — | 2 | 6 | no | — |
| 47 | C2×#127 | — | 2 | 6 | no | — |
| 48 | C4×#41 | D4 | 1 | 1 | yes | characteristic |
| 49 | #509 | — | 2 | 96 | no | — |
| 50 | #508 | — | 2 | 48 | no | — |
| 51 | C2×Q32 | C23 | 1 | 4 | yes | — |
| 52 | C2×SD32 | C23 | 1 | 24 | yes | — |
| 53 | C2×D16 | — | 2 | 48 | no | — |
| 54 | C2×D16 | C23 | 1 | 12 | yes | — |
| 55 | C2×D16 | C23 | 1 | 24 | yes | — |
| 56 | #504 | D4 | 1 | 16 | yes | complemented |
| 57 | C2×M32 | C23 | 1 | 6 | yes | — |
| 58 | C22×C16 | C23 | 1 | 6 | yes | — |
| 59 | #496 | — | 2 | 24 | no | — |
| 60 | #495 | — | 2 | 6 | no | — |
| 61 | #495 | C23 | 1 | 12 | yes | — |
| 62 | #494 | C23 | 1 | 2 | yes | — |
| 63 | #493 | — | 2 | 6 | no | — |
| 64 | #493 | — | 2 | 6 | no | — |
| 65 | #493 | C23 | 1 | 6 | yes | — |
| 66 | #493 | C23 | 1 | 12 | yes | — |
| 67 | #492 | — | 2 | 6 | no | — |
| 68 | #405 | D4 | 1 | 3 | yes | — |
| 69 | #405 | C23 | 1 | 3 | yes | — |
| 70 | C4×M16 | D4 | 1 | 3 | yes | — |
| 71 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 72 | #346 | C23 | 1 | 6 | yes | — |
| 73 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 74 | C4×C16 | C23 | 1 | 3 | yes | — |
| 75 | C4×C16 | C23 | 1 | 6 | yes | — |
| 76 | H(5,𝔽2) | — | 4 | 16 | no | — |
| 77 | H(5,𝔽2) | — | 4 | 16 | no | — |
| 78 | C2×#41 | — | 2 | 12 | no | — |
| 79 | C2×#41 | — | 4 | 4 | no | — |
| 80 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C22×D4 | — | 2 | 12 | no | — |
| 82 | C22×D4 | — | 4 | 12 | no | — |
| 83 | C22×D4 | — | 4 | 12 | no | — |
| 84 | AGL(1,ℤ/8) | — | 4 | 48 | no | — |
| 85 | AGL(1,ℤ/8) | — | 4 | 48 | no | — |
| 86 | #135 | — | 2 | 48 | no | — |
| 87 | #135 | — | 4 | 24 | no | — |
| 88 | C2×Q16 | — | 2 | 2 | no | — |
| 89 | C2×Q16 | C24 | 1 | 4 | yes | — |
| 90 | C2×SD16 | — | 2 | 12 | no | — |
| 91 | C2×SD16 | — | 2 | 12 | no | — |
| 92 | C2×D8 | — | 2 | 48 | no | — |
| 93 | C2×D8 | — | 2 | 6 | no | — |
| 94 | C2×D8 | — | 2 | 12 | no | — |
| 95 | C2×D8 | — | 2 | 12 | no | — |
| 96 | C2×D8 | — | 4 | 24 | no | — |
| 97 | C2×D8 | C24 | 1 | 12 | yes | — |
| 98 | #131 | D8 | 1 | 8 | yes | — |
| 99 | #131 | C2×D4 | 1 | 8 | yes | — |
| 100 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 101 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 102 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 103 | C22×C8 | C24 | 1 | 3 | yes | — |
| 104 | C4⋊Q8 | — | 2 | 2 | no | — |
| 105 | #127 | — | 2 | 6 | no | — |
| 106 | #127 | — | 2 | 12 | no | — |
| 107 | #127 | — | 4 | 12 | no | — |
| 108 | #124 | — | 2 | 12 | no | — |
| 109 | #121 | — | 4 | 12 | no | — |
| 110 | #121 | — | 4 | 24 | no | — |
| 111 | #120 | — | 4 | 24 | no | — |
| 112 | C4×Q8 | D8 | 1 | 2 | yes | — |
| 113 | C4×D4 | — | 4 | 12 | no | — |
| 114 | C4×D4 | D8 | 1 | 6 | yes | — |
| 115 | #117 | C2×D4 | 1 | 3 | yes | — |
| 116 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 117 | Q32 | — | 2 | 8 | no | — |
| 118 | SD32 | — | 2 | 48 | no | — |
| 119 | D16 | — | 2 | 24 | no | — |
| 120 | D16 | — | 2 | 48 | no | — |
| 121 | M32 | C2×D4 | 1 | 24 | yes | complemented |
| 122 | C2×C16 | C2×D4 | 1 | 24 | yes | complemented |
| 123 | C2×C16 | C24 | 1 | 2 | yes | — |
| 124 | C2×C16 | C24 | 1 | 6 | yes | — |
| 125 | #97 | C2×D4 | 1 | 6 | yes | — |
| 126 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 127 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 128 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 129 | C4×C8 | C24 | 1 | 3 | yes | — |
| 130 | C24 | — | 4 | 12 | no | — |
| 131 | #41 | — | 4 | 24 | no | — |
| 132 | #41 | — | 4 | 24 | no | — |
| 133 | #41 | — | 8 | 16 | no | — |
| 134 | #41 | #127 | 1 | 4 | yes | — |
| 135 | #41 | C2×D8 | 1 | 4 | yes | — |
| 136 | C2×Q8 | — | 2 | 4 | no | — |
| 137 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 138 | C2×D4 | — | 2 | 12 | no | — |
| 139 | C2×D4 | — | 4 | 12 | no | — |
| 140 | C2×D4 | — | 4 | 24 | no | — |
| 141 | C2×D4 | — | 4 | 24 | no | — |
| 142 | C2×D4 | — | 4 | 12 | no | — |
| 143 | C2×D4 | — | 4 | 12 | no | — |
| 144 | C2×D4 | — | 4 | 24 | no | — |
| 145 | C2×D4 | — | 8 | 48 | no | — |
| 146 | C2×D4 | — | 16 | 24 | no | — |
| 147 | C2×D4 | C2×D8 | 1 | 3 | yes | — |
| 148 | C22×C4 | — | 4 | 12 | no | — |
| 149 | C22×C4 | C2×D8 | 1 | 6 | yes | — |
| 150 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 151 | Q16 | — | 2 | 8 | no | — |
| 152 | Q16 | — | 4 | 4 | no | — |
| 153 | SD16 | — | 4 | 24 | no | — |
| 154 | SD16 | — | 4 | 24 | no | — |
| 155 | D8 | — | 2 | 24 | no | — |
| 156 | D8 | — | 4 | 24 | no | — |
| 157 | D8 | — | 4 | 12 | no | — |
| 158 | D8 | — | 4 | 24 | no | — |
| 159 | M16 | #127 | 1 | 12 | yes | — |
| 160 | M16 | C2×D8 | 1 | 12 | yes | — |
| 161 | C2×C8 | C2×D8 | 1 | 12 | yes | — |
| 162 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 163 | C2×C8 | C22×D4 | 1 | 12 | yes | — |
| 164 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 165 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 166 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 167 | C4⋊C4 | — | 4 | 4 | no | — |
| 168 | C4⋊C4 | C2×D8 | 1 | 6 | yes | — |
| 169 | C22⋊C4 | — | 4 | 24 | no | — |
| 170 | C22⋊C4 | C2×D8 | 1 | 6 | yes | — |
| 171 | C42 | C2×D8 | 1 | 6 | yes | — |
| 172 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 173 | C42 | C22×D4 | 1 | 3 | yes | — |
| 174 | C16 | C22×D4 | 1 | 4 | yes | — |
| 175 | C16 | C22×D4 | 1 | 12 | yes | — |
| 176 | C23 | — | 4 | 12 | no | — |
| 177 | C23 | — | 8 | 24 | no | — |
| 178 | C23 | — | 16 | 24 | no | — |
| 179 | C23 | C22×D8 | 1 | 3 | yes | — |
| 180 | Q8 | — | 4 | 4 | no | — |
| 181 | Q8 | — | 4 | 4 | no | — |
| 182 | Q8 | #493 | 1 | 4 | yes | — |
| 183 | D4 | — | 4 | 12 | no | — |
| 184 | D4 | — | 4 | 12 | no | — |
| 185 | D4 | — | 8 | 24 | no | — |
| 186 | D4 | — | 8 | 24 | no | — |
| 187 | D4 | — | 8 | 24 | no | — |
| 188 | D4 | — | 16 | 24 | no | — |
| 189 | D4 | #493 | 1 | 12 | yes | — |
| 190 | C2×C4 | — | 2 | 12 | no | — |
| 191 | C2×C4 | — | 4 | 4 | no | — |
| 192 | C2×C4 | — | 8 | 24 | no | — |
| 193 | C2×C4 | C2×#127 | 1 | 6 | yes | — |
| 194 | C2×C4 | C22×D8 | 1 | 6 | yes | — |
| 195 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 196 | C2×C4 | C22×D8 | 1 | 6 | yes | — |
| 197 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 198 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 199 | C8 | C2×#127 | 1 | 6 | yes | — |
| 200 | C8 | C22×D8 | 1 | 2 | yes | — |
| 201 | C8 | C22×D8 | 1 | 6 | yes | — |
| 202 | C8 | C23×D4 | 1 | 2 | yes | — |
| 203 | C22 | — | 2 | 6 | no | — |
| 204 | C22 | — | 8 | 24 | no | — |
| 205 | C22 | — | 8 | 12 | no | — |
| 206 | C22 | — | 32 | 24 | no | — |
| 207 | C22 | C2×#493 | 1 | 6 | yes | — |
| 208 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 209 | C4 | — | 2 | 6 | no | — |
| 210 | C4 | — | 8 | 4 | no | — |
| 211 | C4 | C2×#493 | 1 | 6 | yes | — |
| 212 | C4 | #3415 | 1 | 4 | yes | complemented |
| 213 | C4 | C22×#127 | 1 | 1 | yes | characteristic |
| 214 | C4 | C23×D8 | 1 | 1 | yes | characteristic |
| 215 | C2 | — | 2 | 6 | no | — |
| 216 | C2 | — | 16 | 12 | no | — |
| 217 | C2 | C22×#493 | 1 | 1 | yes | characteristic |
| 218 | C2 | C2×#3415 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta